Chapter 6
Specification:
Choosing the
Independent
Variables
Copyright © 2011 Pearson Addison-Wesley.
All rights reserved.
Slides by Niels-Hugo Blunch
Washington and Lee University
Specifying an Econometric
Equation and Specification Error
• Before any equation can be estimated, it must be completely
specified
• Specifying an econometric equation consists of three parts,
namely choosing the correct:
– independent variables
– functional form
– form of the stochastic error term
• Again, this is part of the first classical assumption from Chapter 4
• A specification error results when one of these choices is made
incorrectly
• This chapter will deal with the first of these choices (the two other
choices will be discussed in subsequent chapters)
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6-1
Omitted Variables
• Two reasons why an important explanatory variable
might have been left out:
– we forgot…
– it is not available in the dataset, we are examining
• Either way, this may lead to omitted variable bias
(or, more generally, specification bias)
• The reason for this is that when a variable is not
included, it cannot be held constant
• Omitting a relevant variable usually is evidence that the
entire equation is a suspect, because of the likely bias of
the coefficients.
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6-2
The Consequences of an
Omitted Variable
•
Suppose the true regression model is:
(6.1)
Where
•
is a classical error term
If X2 is omitted, the equation becomes instead:
(6.2)
Where:
(6.3)
•
Hence, the explanatory variables in the estimated regression (6.2) are not
independent of the error term (unless the omitted variable is uncorrelated
with all the included variables—something which is very unlikely)
•
But this violates Classical Assumption III!
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6-3
The Consequences of an Omitted
Variable (cont.)
• What happens if we estimate Equation 6.2 when Equation 6.1 is the truth?
• We get bias!
• What this means is that:
(6.4)
• The amount of bias is a function of the impact of the omitted variable on the
dependent variable times a function of the correlation between the included
and the omitted variable
• Or, more formally:
(6.7)
• So, the bias exists unless:
1. the true coefficient equals zero, or
2. the included and omitted variables are uncorrelated
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6-4
Correcting for an Omitted
Variable
• In theory, the solution to a problem of specification bias seems easy:
add the omitted variable to the equation!
• Unfortunately, that’s easier said than done, for a couple of reasons
1. Omitted variable bias is hard to detect: the amount of bias introduced can
be small and not immediately detectable
2. Even if it has been decided that a given equation is suffering from omitted
variable bias, how to decide exactly which variable to include?
© 2011 Pearson Addison-Wesley. All rights reserved.
6-5
Correcting for an Omitted
Variable (cont.)
• What if:
– You have an unexpected result, which leads you to believe that you have
an omitted variable
– You have two or more theoretically sound explanatory variables as
potential “candidates” for inclusion as the omitted variable to the equation is
to use
• How do you choose between these variables?
• One possibility is expected bias analysis
– Expected bias: the likely bias that omitting a particular variable would have
caused in the estimated coefficient of one of the included variables
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6-6
Correcting for an Omitted
Variable (cont.)
• Expected bias can be estimated with Equation 6.7:
(6.7)
• When do we have a viable candidate?
– When the sign of the expected bias is the same as the sign
of the unexpected result
• Similarly, when these signs differ, the variable is
extremely unlikely to have caused the unexpected
result
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6-7
Irrelevant Variables
• This refers to the case of including a variable in an equation when it
does not belong there
• This is the opposite of the omitted variables case—and so the impact
can be illustrated using the same model
• Assume that the true regression specification is:
(6.10)
• But the researcher for some reason includes an extra variable:
(6.11)
• The misspecified equation’s error term then becomes:
(6.12)
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6-8
Irrelevant Variables (cont.)
• So, the inclusion of an irrelevant variable will not cause bias
(since the true coefficient of the irrelevant variable is zero, and so
the second term will drop out of Equation 6.12)
• However, the inclusion of an irrelevant variable will:
– Increase the variance of the estimated coefficients, and this
increased variance will tend to decrease the absolute
magnitude of their t-scores
– Decrease the R2 (but not the R2)
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6-9
Four Important Specification
Criteria
• We can summarize the previous discussion into four criteria to help
decide whether a given variable belongs in the equation:
1. Theory: Is the variable’s place in the equation unambiguous and theoretically
sound?
2. t-Test: Is the variable’s estimated coefficient significant in the expected direction?
3. R2: Does the overall fit of the equation (adjusted for degrees of freedom) improve
when the variable is added to the equation?
4. Bias: Do other variables’ coefficients change significantly when the variable is
added to the equation?
• If all these conditions hold, the variable belongs in the equation
• If none of them hold, it does not belong
• The tricky part is the intermediate cases: use sound judgment!
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6-10
Specification Searches
• Almost any result can be obtained from a given
dataset, by simply specifying different regressions until
estimates with the desired properties are obtained
• Hence, the integrity of all empirical work is open to
question
• To counter this, the following three points of Best
Practices in Specification Searches are suggested:
1. Rely on theory rather than statistical fit as much as possible when
choosing variables, functional forms, and the like
2. Minimize the number of equations estimated (except for
sensitivity analysis
3. Reveal, in a footnote or appendix, all alternative
specifications estimated
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6-11
Bias Caused by Relying on the
t-Test to Choose Variables
•
Dropping variables solely based on low t-statistics may lead to two
different types of errors:
1. An irrelevant explanatory variable may sometimes be included in the
equation (i.e., when it does not belong there)
2. A relevant explanatory variables may sometimes be dropped from the
equation (i.e., when it does belong)
•
In the first case, there is no bias but in the second case there is bias
•
Hence, the estimated coefficients will be biased every time an excluded
variable belongs in the equation, and that excluded variable will be left out
every time its estimated coefficient is not statistically significantly different
from zero
•
So, we will have systematic bias in our equation!
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6-12
Sensitivity Analysis
• Contrary to the advice of estimating as few equations as possible
(and based on theory, rather than fit!), sometimes we see journal article
authors listing results from five or more specifications
• What’s going on here:
• In almost every case, these authors have employed a technique called
sensitivity analysis
• This essentially consists of purposely running a number of alternative
specifications to determine whether particular results are robust (not
statistical flukes) to a change in specification
• Why is this useful? Because true specification isn’t known!
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6-13
Data Mining
• DANGER: Data mining involves exploring a data set to try
to uncover empirical regularities that can inform
economic theory
• That is, the role of data mining is opposite that of traditional
econometrics, which instead tests the economic theory on
a data set
• Be careful, however!
– a hypothesis developed using data mining techniques must be
tested on a different data set (or in a different context) than
the one used to develop the hypothesis
– Not doing so would be highly unethical: After all, the researcher
already knows ahead of time what the results will be!
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6-14
Key Terms from Chapter 6
• Omitted variable
• Irrelevant variable
• Specification bias
• Sequential specification search
• Specification error
• The four specification criteria
• Expected bias
• Sensitivity analysis
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6-15